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Homogeneous Nucleation and the Spinodal Line

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Homogeneous Nucleation and the Spinodal Line Reprinted from JOURNAL OF HEAT TRANSFER, Vol. 103, No.1 , February 1981 J. H. Lienhard1 Homogeneous Nucleation and the Professor, Mechanical Engineering Department, UnIversity of Houston, 2 Houston, lex 77004 Spinodal Line Fellow ASME Amir Karimi The limit of homogeneous nucleation in a liquid is shown to lie very close to its liquid spi­ Research Assistant, nodal line. It is also argued that the homogeneous nucleation prediction should be based BOiling and Phase Change Laboratory, on a comparison of the critical work of nucleation with the "potential well" energy instead Mechanical Engineering Department, of the kinetic molecular energy. The result is a new prediction of the liquid spinodal line University of Kentucky, for water that is valid to large negative pressures. This prediction compares well with spi­ Lexington, KY 40506 Student Mem. ASME nodal points obtained by extrapolating liquid and vapor water data with the Himpan equation. Introduction How Close to the Spinodal Line Can a Fluid Be Objective. There have been some problems in the literature as to Brought? the relation between the homogeneous nucleation limit and the spi­ The minimum work required to bring a fluid from a homogeneous nodal line. Indeed, when we [1] argued that the two were nearly the nucleation temperature, Tn, to the spinodal temperature, T s, at the same in a superheated liquid, but quite different in a subcooled vapor, same pressure, Pn, is given by the change of the thermodynamic we were questioned sharply by referees. Some of them objected that availability between the two points, !::.a. the spinodal line was a fictional limit that could not ever be demon­ !::.a = (h - h ) - Tn (ss - Sn) (3) strated experimentally. s n At the time, our evidence that the two liquid limits were nearly the where the reference, or dead state, is specified as the pressure, Pn, and same was strong, but only circumstantial. A direct argument must still the initial temperature, Tn. Figure 1 shows this hypothetical process. be offered to show that the two limits lie close to one another. Such Notice that we arbitrarily consider an isobaric process. If there exists a demonstration would have considerable value because the spinodal a path requiring less energy, then the calculation based on the isobaric line- not the homogeneous limit-defines the local minimum in a model will be conservative. correct equation of state. But only the latter can be established ex­ Since the process from point (n) to point (s) is isobaric, equation perimentally. If the two are close, then the spinodal line will be known (3) becomes as well. We would then have another constraint to add in the formu­ lation of an equation of state that will have validity in the metastable !::.a = fT~ ' (c p - ; Cp ) dT, (4) liquid regime. The Spinodal Limit. Conventional equations of state are written and the problem of evaluating the minimum work reduces to that of to describe the gas and liquid states as though they were continua­ specifying cp(T) in the neighborhood of the spinodal line. We know not made of molecules. Since such equations must satisfy the Max­ from elementary thermodynamic considerations that well-Gibbs requirement that Limit Limit [J:' T, J: T, C dT] g. sat vapor cp(T) = 00; cpdT and - p- = finite (5) dg = i vdp = 0 (1) T-~ T-~ T T T igl sat liquid Among the functions that satisfy these conditions are: cp ~ (Ts - T)-b we expect their isotherms to pass through ridges or "spines" of local where 1 > b > 0, and cp ~ In(Ts - T). So too is any cp that approaches maxima and minima defined by infinity at Ts as a power weaker than a linear function of (T - Ts ). We can clearly form an upper bound on !::.a by factoring out the ap I = 0 (2) largest value of (T - Tn)/T. Thus av T Since the regions between these spines (or "spinodal lines" as they Ts - Tn J: T, !::.aupper bound'" !::.a" =--- cpdT (6) are called) is unstable, the state of real f1uids cannot be brought all T s Tn the way to either spinodal line. Molecular fluctuations will inevitably Substituting anyone of the acceptable cp functions in equation (6) destabilize the f1uid before the spine of instability is reached. we obtain We subscribe to the view that the spinodal limit is nevertheless a (Ts - Tn)2 physically useful concept in that the equation of state of a real f1uid !::.a,,=D cp(T n) (7) must satisfy both equations (1) and (2). Furthermore, we have pre­ T s viously shown [1, 2J that observations of homogeneous nucleation in where D is a number larger than unity. The value of !::.a" per molecule liquids probably reach values quite close to the spinodal line. By is then !::.a u/N A. providing an analytical demonstration that this is true, we will make The conventional homogeneous nucleation theory says that nu­ it possible to specify the liquid spinodal in an equation of state for cleation is virtually sure to occur when the critical work required to water. This is a key step in a program that we have undertaken (see trigger nucleation is on the order of magnitude of lOkTn per nucleus.3 [3]) to develop a fundamental equation that can be used in the met­ It follows that astable regimes. 10kTn D cp(Tn) (Ts - Tn)2 --< --"'-'--"-- ---'-----"'- (8) N n NA Ts where N n is the number of liquid molecules in the region displaced by a nucleus bubble. Then 1 This work was done while the first author was at the University of Ken­ (Ts - Tn)2 10_R_~ (9) tucky. > 2 This work was done under the support of the Electric Power Research Inst. TnTs D cp(Tn) N n (EPRI Contract RP 678·1) with BalIaj Sehgal as project manager. Contributed by the Heat Transfer Division for publication in the JOURNAL OF HEAT TRANSFER Manuscript received by the Heat Transfer Division July 15, 1980. 3 The background for this assertion is developed in the next section. Journal of Heat Transfer FEBRUARY 1981, VOL. 103 / 61 J Nucleation events . nucleation events m3s Gb liquid spinodal line ---------= e- (Il) IA"', } molecule-collision molecules collisions N B--­ /' \ "" m 3 s " where the Gibbs number, Gb, is a. critical work to trigger nucleation -T----- -- Ts Gb = (12) ~ Psot :0 kT <.f) vap'or ",, Tn <.f) The value of j at which nucleation absolutely must occur is the largest ~ \- spinodal " a. line " possible one. It corresponds with the minimwn possible value of the "'­ critical work. "­ "- One way of specifying the maximum possible value of j is to imagine Pn Minimum work from n to s that just one nucleation event occurs every relaxation-time within the population of liquid displaced by the nucleus bubble of radius, re. : 6al~ Thus, volume, v collisions ] [ s ] Fig. 1 Process 01 triggering nucleation Irom the point 01 homogeneous nu­ } - [relaxation time B collisions cleation, n, 10 the point, s, on Ihe spinodal line at the same pressure where the molar ideal gas constant, R " kNA· Consider next the order of magnitude of the three factors on the 3 x [l:!-.- gm m ] (l3) left side of equation (9): The term (lOID) < 10. R/c p clearly ap­ NA molecules Pf gm proaches zero at the spinodal line, but we do not yet know how close This calculation and the experimental data of Skripov, et at. (see, e.g., Tn is to Ts. However, for water at 1 atm, R is 0.46 kJ/kg - K while c p [5].) can be interpreted to give somewhat varying results, but a good is 4.2 kJ/kg - K at saturation and at least 2.5 times this value at Tn upper bound appears to be in the neighborhood of: (see, e.g., [3]). Thus for water, R/cp < 0.04 and for other liquids it will be much less. Finally, Skripov [4] has calculated N n for a variety of j "" (10)-5 or Gb "" 11.5 (14) organic f1uids at high pressure. He obtained numbers between 270 and 1010 at his observed nucleation temperatures. For water at 1 atm, This corresponds withJ "" 1034 m-3 S-I. N n increases to 4000. The prediction of the spinodal line is then completed by substi­ It follows that for water at 1 atm, (Ts - T n)/Tn is substantially less tuting this value of j in equation (11), using Frenkel's expression than 0.01 and for organic substances the result should be still less [6] owing to far smaller values of . Then in general R/c p critical work = 47rre 2cr/ 3 (15) (10) and a critical nucleus given by [4J If a comparable argument were developed for the vapor phase re = 2cr/[ (1 - Uf/ Ug )(PSQt - Pn) J (16) spinodal line, an equation similar to equation (9) would result. But in this case, N n can be very small because there are very few vapor Such predictions have frequently been offered in the past to predict molecules within a volume equal to that occupied by a nucleus homogeneous nucleation and they have worked fairly well at the drop. spinodal temperatures that occur at positive pressures. These tem­ Equation (10) will therefore no longer be true. Nucleation thus peratures are usually in the range: 0.9 <' Ts/Tc < 1.0 (see, e.g., [4]). occurs very close to the liq uid spinodal, but we cannot expect it to occur anywhere near the vapor spinodal. This is exactly what we A Modification of the Conventional Theory showed previously with experimental data in [1].
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